Question

Suppose a firm's production function is given by the equation  Q = 12L^.5K^.5 . This firm operates in the short run where capital (K) is fixed at a quantity of 16. If the price per unit of the good is $1.8 and labor costs $10 per unit. Then the profit maximizing amount of labor in the short run is ___?

Answer 1

Answer

Total Cost = wL + vK where w = wage rate = 10 , v = rent(and is constant)

=> Total Cost = 10L + 16v

Total revenue = Price *Quantity = 1.8Q = 1.8*12L^.5K^.5 = 1.8*12L^.516^.5 = 86.4L^.5

Profit(Pr) = Total revenue - Total Cost

=> Profit(Pr) = 86.4L^.5 - 10L + 16v

First order condition;

d(Pr)/dL = 0 => 0.5*86.4/L^.5 - 10 = 0

=> L = (0.5*86.4/10)²

=> L = 18.66

Thus, the profit maximizing amount of labor in the short run is 18.66 or 19 (approx) units